We sample some Poincaré-Birkhoff-Witt theorems appearing in mathematics. Along the way, we compare modern techniques used to establish such results, for example, the Composition-Diamond Lemma, Gröbner basis theory, and the homological approaches of Braverman and Gaitsgory and of Polishchuk and Positselski. We discuss several contexts for PBW theorems and their applications, such as Drinfeld-Jim...

For a symmetrizable Kac-Moody algebra the category of admissible representations is an analogue of the category of finite dimensional representations of a semisimple Lie algebra. The monoid associated to this category and the category of restricted duals by a generalized Tannaka-Krein reconstruction contains the Kac-Moody group as open dense unit group and has similar properties as a reductive ...

The major aim of this research is benchmarking Honeywell Federal Manufacturing &Technologies projects using Data Envelopment Analysis (DEA) technique and to compare the results with that of published recently in Qing Cao, James J. Hoffman Qing Cao, James J. Hoffman. (2011). A case study approach for developing a project performance evaluation system. International Journal of Project Managem...

Notions of orthogonality and convolution are explored with the use Kontorovich–Lebedev transform its convolution. New classes corresponding orthogonal polynomials functions investigated. Integral representations, relations explicit expressions established.

in this paper we introduce two-wavelet constants for square integrable representations of homogeneous spaces. we establishthe orthogonality relations for square integrable representationsof homogeneous spaces which give rise to the existence of aunique self adjoint positive operator on the set of admissiblewavelets. finally, we show that this operator is a constant multiple of identity operator...

‎From Hilbert's theorem of zeroes‎, ‎and from Noether's ideal theory‎, ‎Birkhoff derived certain algebraic concepts (as explained by Tholen) that have a dual significance in general toposes‎, ‎similar to their role in the original examples of algebraic geometry‎. ‎I will describe a simple example that illustrates some of the aspects of this relationship‎. The dualization from algebra to geometr...

An algebra $mathbf A = langle A, to, 0 rangle$, where $to$ is binary and $0$ a constant, called an implication zroupoid ($mathcal{I}$-zroupoid, for short) if $mathbf{A}$ satisfies the identities: $(x to y) z approx [(z' x) (y z)']'$, $x' : x 0$, $ 0'' 0$. These algebras generalize De Morgan $vee$-semilattices with zero. Let $mathcal{I}$ denote variety of zroupoids. The investigations into struc...

We present and prove in detail a Poincaré–Birkhoff–Witt commutator lemma for the quantum superalgebra Uq[gl(m|n)].

In this paper we prove that the periodic Toda lattice admits globally defined Birkhoff coordinates.